DCluster {DCluster}

A Package for the Detection of Spatial Clusters of Diseases for Count Data

Description

DCluster is a collection of several methods related to the detection of spatial clusters of diseases. Many widely used methods, such as Openshaw's GAM, Besag and Newell, Kulldorff and Nagarwalla, and others have been implemented.

Besides the calculation of these statistic, bootstrap can be used to test its departure from the null hypotheses, which will be no clustering in the study area. For possible sampling methods can be used to perform the simulations: permutation, Multinomial, Poisson and Poisson-Gamma.

Minor modifications have been made to the methods to use standardized expected number of cases instead of population, since it provides a better approach to the expected number of cases.

Introduction

We'll always suppose that we are working on a study region which is divided into n non-overlaping smaller areas where data are measured. Data measured are usually people suffering from a disease or even deaths. This will be refered as Observed number of cases. For a given area, its observed number of cases will be denoted by O_i and the sum of these quantities over the whole study region will be O_+.

In the same way can be defined Population and Standardized Expected number of cases, which will be denoted by P_i and E_i, respectively. The sum of all these quantities are represented by P_+ and E_+.

The basic assumption for the data is that they are independant observations from a Poisson distribution, whose mean is \theta_iE_i, where \theta_i is the relative risk. That is,

O_i \sim Po(\theta_i E_i); \ i=1, \ldots , n

Null hypotheses

Null hypotheses is usually equal relative risks, that is

H_0: \theta_1= \ldots = \theta_n = \lambda

\lambda may be considered to be known (one, which means standard risk) or unknown. In the last case, E_i must slightly be corrected by multiplying it by the overall relative risk \frac{O_+}{E_+}.

Code structure

Function names follow a common format, which is a follows:

method name.stat

Calculate the statistic itself.

method name.boot

Perform a non-parametric bootstrap.

method name.pboot

Perform a parametric bootstrap.

Openshaw's G.A.M. has generally been implemented in a function called gam, which some methods ( Kulldorff & Nagarwalla, Besag & Newell) also use, since they are based on a window scan of the whole region. At every point of the grid, a function is called to determine whether that point is a cluster or not. The name of this function is shorten method name.iscluster.

This function calculates the local value of the statistic involved and its signifiance by means of bootstrap. The interface provided, through function gam, is quite straightforward to use and it can handle the three methods mentioned and other supplied by the users.

Bootstrap procedures

Four possible bootstrap models have been provided in order to estimate sampling distributions of the statistics provided. The first one is a non-parametric bootstrap, which performs permutations over the observed number of cases, while the three others are parametric bootstrap based on Multinomial, Poisson and Poisson-Gamma distributions.

Permutation method just takes observed number of cases and permute them among all regions, to know whether risk in uniform across the whole study area. It just should be used with care since we'll face the problem of having more observed cases than population in very small populated areas.

Multinomial sampling is based on conditioning the Poisson framework to O_+. THis way (O_1, \ldots, O_n) follows a multinomial distribution of size O_+ and probabilities (\frac{E_1}{E_+}, \ldots, \frac{E_n}{E_+}).

Poisson sampling just generates observed number of cases from a Poisson distribution whose mean is E_i.

Poisson-Gamma sampling is based on the Poisson-Gamma model proposed by Clayton and Kaldor (1984):

O_i|\theta_i \sim Po(\theta_i E_i)

\theta_i \sim Ga(\nu, \alpha)

The distribution of O_i unconditioned to \theta_i is Negative Binomial with size \nu and probability \frac{\alpha}{\alpha+E_i}. The two parameters can be estimated using an Empirical Bayes approach from the Expected and Observed number of cases. Function empbaysmooth is provided for this purpose.

Data

One of the parameters, which is usually called data, passed to many of the functions in this package is a dataframe which contains the data for each of the regions used in the analysis. Besides, its columns must be labeled:

Observed

Observed number of cases.

Expected

Standardised expected number of cases.

Population

Population at risk.

x

Easting coordinate of the region centroid.

y

Northing coordinate of the region centroid.

References

Clayton, David and Kaldor, John (1987). Empirical Bayes Estimates of Age-standardized Relative Risks for Use in Disease Mapping. Biometrics 43, 671-681.

Lawson et al (eds.) (1999). Disease Mapping and Risk Assessment for Public Health. John Wiley and Sons, Inc.

Lawson, A. B. (2001). Statistical Methods in Spatial Epidemiology. John Wiley and Sons, Inc.